Iteration properties

Every monoid element has two sets associated with it that define the properties of its iteration:

• Iteration set: this define the set of numbers with which this function can be iterated over. Injective functions include -1 as in their iteration set.
• Iteration equivalence relation: this defines equivalent iteration numbers, for example, all iterations other then zero are equal to one with idempotent elements.

These sets can be described as a single partition of the real numbers that includes a specially flagged set of invalid values. The iteration properties of an element form a distributive lattice, and the meet operation in this lattice describes the iteration properties of the composition of several elements.